Graphical ModelsThe idea of modelling systems using graph theory has its origin in several scientific areas: in statistical physics (the study of large particle systems), in genetics (studying inheritable properties of natural species), and in interactions in contingency tables. The use of graphical models in statistics has increased considerably over recent years and the theory has been greatly developed and extended. This book provides the first comprehensive and authoritative account of the theory of graphical models and is written by a leading expert in the field. It contains the fundamental graph theory required and a thorough study of Markov properties associated with various type of graphs. The statistical theory of log-linear and graphical models for contingency tables, covariance selection models, and graphical models with mixed discrete-continous variables in developed detail. Special topics, such as the application of graphical models to probabilistic expert systems, are described briefly, and appendices give details of the multivarate normal distribution and of the theory of regular exponential families. The author has recently been awarded the RSS Guy Medal in Silver 1996 for his innovative contributions to statistical theory and practice, and especially for his work on graphical models. |
Contents
| 1 | |
Subgraphs of decomposable graphs | 13 |
4 | 29 |
19 | 45 |
Contingency tables | 64 |
Notation | 123 |
Bibliographical notes | 129 |
1 | 158 |
Further topics | 222 |
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Common terms and phrases
algorithm assume asymptotic Biometrika block-recursive calculated canonical statistic CG density CG distribution chain components computation conditional distribution conditional independence consider contingency tables corresponding covariance matrix covariance selection model decomposable graphs decomposable models decomposition denote direct join directed acyclic graph edge equivalent example factorization follows given global Markov graph G graphical models H₁ Hence hierarchical model homogeneous hypergraphs interaction models inverse iterative Jensen Journal Lauritzen Lemma likelihood function likelihood ratio linear log-affine model log-linear models marginal table Markov property maximizing maximum likelihood estimate model with graph multinomial sampling multivariate normal n(ia normal distribution obtained pairwise partitioned perfect sequence positive definite probability Proof Proposition quadratic random variables random vector recursive regular exponential model restrictions result Royal Statistical Society satisfies saturated model Section space subgraph subsets subspace Theorem triangulated undirected graph vertex vertices Wermuth Wishart distribution zero ΕΙ